How can I derive a relationship between temperature and coefficient of restitution for a rubber bouncy ball theoretically? For a high school research project, I am exploring the relationship between temperature and coefficient of restitution for a rubber bouncy ball. I have already carried out the experiment and have formed a relationship practically. However, I would also like to derive a theoretical formula for my hypothesis which I can then use to calculate the theoretical value of coefficient of restitution for the rubber ball at different temperatures, post which I can compare this with the values I obtained experimentally.
I carried out the experiment in terms of height and therefore used the below formula to calculate COR at each temperature:
$$\text{Coef. of Rest}=\sqrt{\frac{\text{Height }R}{\text{Height }D}}$$
where HeightR is the rebound height upon the first bounce and HeightD is the height the ball was dropped from.
I would appreciate any guidance on how I should derive an equation to relate coefficient of restitution to temperature for a rubber bouncy ball theoretically so that I progress in the correct direction. Just to clarify, I want to derive an equation for coefficient of restitution in terms of the temperature which the ball is at.
 A: 
I want to derive an equation for coefficient of restitution in terms
of the temperature which the ball is at.

That's a very BIG ask.
The CoR is related to the concept of Elastic Hysteresis:

Elastic Hysteresis is the difference between the strain energy
required to generate a given stress in a material, and the material's
elastic energy at that stress. This energy is dissipated as internal
friction (heat) in a material during one cycle of testing (loading and
unloading).
When mechanical test data is plotted on a stress/strain curve, a
material exhibiting elastic hysteresis will display one path during
the loading phase of the test, and a different path during the
unloading phase. The two paths will clearly diverge due to hysteresis
loss (energy loss in the form of heat), with the area between the
curves representing the energy dissipated.

Graphically:

Now you face two main problems.

*

*Relation between simple 'lab' hysteresis and complex 'real life' hysteresis:
The Elastic Hysteresis of a visco-elastic material (like rubber) can be established using simple geometries like standardised dumbbells, where very simple deformations are in play but to try and predict the hysteresis of something as complex as a bouncing ball from these simple lab data is a different matter altogether.


*Influence of temperature on Elastic Hysteresis:
A theoretical model linking these two variables is not available as far as I know.
Some relevant research has almost certainly been carried out in the field of (automotive) timing belts, as the energy losses due to hysteresis is an area of concern in that field of application. So it might be an idea to trawl the specialised trade literature for some insights.

So rather than try and develop a complex and potentially quite faulty theoretical relation between the CoR and temperature it might be much preferable to develop an empirical model.
A: From the coefficient of restitution you can calculate the loss of the macroscopic kinetic energy of the ball, macroscopic meaning the loss of external kinetic energy of the ball as a whole ($1/2mv^2$).
Then, neglecting sound and any possible inelastic deformation of the surface on which the ball bounces, you can assume all of the loss macroscopic kinetic energy equals the increase in microscopic kinetic energy of the molecules of the ball, its internal kinetic energy, which results in an increase in the temperature of the ball.
Finally, knowing the mass of the ball and the specific heat $C$ of its material, you can calculate the theoretical increase in temperature from
$$\frac{1}{2}mv^{2}=mC\Delta T$$
Where the left side of the equation is the loss of KE of the ball.
The theoretical temperature increase is likely to be small and not uniform throughout the ball. Moreover, the increase in temperature relative to the environment will result in heat transfer to the lower temperature environment. In short, don't expect to be able to actually measure the temperature increase, or at least don't expect to be able to measure it accurately.
Hope this helps.
